Some Simple Algorithms for the Evaluations and Representations of the Riemann Zeta Function at Positive Integer Arguments

Many interesting solutions of the so-called Basler problem of evaluating the Riemann zeta function ~r(s) when s = 2, which was of vital importance to Euler and the Bernoulli brothers (Jakob and Johann Bernoulli), have appeared in the mathematical literature ever since Euler first solved this problem in the year 1736. The main object of the present paper is to investigate rather systematically several interesting evaluations and representations of st(s) when s e I%1\ {1}. In one of many computationally useful special cases considered here, it is observed that ~r(3) can be represented by means of a series which converges much more rapidly than that in Eulerʹs celebrated formula as well as the series used recently by Ap6ry in his proof of the irrationality of ~ʹ(3). Symbolic and numerical computations using Mathematica (Version 4.0) for Linux show, among other things, that only 50 terms of this series are capable of producing an accuracy of seven decimal places.